Wondering if I made things worse when I tried to apologize to a sister

Assalamu alaikum. I think I might have accidentally been a bit rude to a sister (she was wearing a colourful hijab). She certainly felt I was. To make it right, I tried to apologize, explaining my harsh words weren't directed at her, and I instinctively reached out my hand. She didn't accept the apology and said I was still being aggressive. It really stayed with me, and the next day I remembered something: a former neighbor once told me she avoids shaking hands with non-mahram men for religious reasons. I'm now wondering, did my attempt to apologize by offering a handshake actually make the situation worse? Could someone with knowledge help me understand this better? JazakAllah khair.